# PhD opportunities

## Research Environment

The Integrable Systems group has its own seminar series and frequently hosts Workshops and Conferences in Leeds. It has national and international links and collaborative networks. Some of our previous PhD students and post-doctoral researchers contributed lectures in the LMS Classical and Quantum Integrability programme.

Below is a selection of PhD topics proposed in our group. For more information about these, feel free to contact the designated supervisor directly.

## Funding and Admissions Details

Prospective students are welcome to e-mail any staff member of the group to discuss research possibilities. Further details of possible funding and of how to apply can be found on the Postgraduate research opportunities page.

## List of PhD topics:

### Lagrangian multiforms and quantisation of integrable systems

### Supervisors: Dr Vincent Caudrelier

**Description**: Lagrangian multiforms were introduced in 2009 by S. Lobb and F. Nijhoff here in Leeds as a variational framework to describe integrability of integrable systems in classical mechanics and field theory. The new idea is to formulate a generalised variational principle for a collection of Lagrangians assembled in a Lagrangian multiform. The resulting collection of Euler-Lagrange equations has the new important feature that it provides equations on the allowed Lagrangians, in addition to the usual Euler-Lagrange equations giving the equations of motion.

The theory has been developed steadily since its introduction then and it is now rather well understood how it relates to the traditional approaches to integrability such as Hamiltonian and Lax pair formulations. A tantalising open direction of research is the possibility of using Lagrangian multiforms to quantize integrable systems following Feynman's idea of path integrals.

This PhD project proposes to investigate this quantization problem by taking advantage of the integrability of the theory encoded in the multiform. Since multiforms apply equally well for finite or infinite dimensional systems, we expect that one should be able to describe quantum mechanical systems and quantum field theories for which the classical Lagrangian multiform is known. Current examples abound and include: the Toda chain, Gaudin models, the sine-Gordon equation, the modified Korteveg-de Vries equation, the nonlinear Schrödinger equation, Zakharov-Mikhailov models which contain

the Faddeev-Reshetikhin model and recently introduced deformed sigma/Gross-Neveu models as particular cases, etc. Possible tools of investigation can include discretisation of the path integral, as in Feynman's original work, or equivariant localisation techniques. An alternative and complementary direction would be to develop covariant canonical quantisation based on results obtained by the supervisor on covariant Poisson brackets and classical r-matrix structures. Comparison with, as well as guidance from, the well-established theory of the quantum Yang-Baxter equation and quantum groups will be important components of the project.

### Quantum Variational Principle and Discrete Integrable Systems

### Supervisor: Professor Frank Nijhoff

**Description**: The project deals with a novel quantum theory that is based on the mathematical structure behind so-called integrable systems, i.e., models that on the classical level are described by certain (mostly nonlinear) differential or difference equations and that exhibit the aspect of `exact solvability' - the fact that they are amenable to exact (rather than approximate or numerical) methods for their solution. Such systems possess a rich mathematical structure (e.g. connections to infinite-dimensional Lie algebras and quantum groups). A key feature is the notion of multidimensional consistency, the property that such models can be extended to compatible systems of equations in spaces of arbitrary dimension. In 2009 Lobb and Nijhoff proposed a novel variational (i.e., least-action) principle that describes this remarkable feature within a Lagrangian formalism.

The new principle forms potentially a paradigm for a new type of fundamental physics.

Recently, the ideas behind this new approach were extended to the quantum realm, and the first steps were taken to formulate a quantum version of this variational principle. The project for a PhD student is to continue this work into the quantum variational principle, elaborating various (integrable) model systems aimed at building a coherent framework together with the development of some new mathematical methodologies.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.

### Elliptic Discrete Integrable Systems

### Supervisor: Professor Frank Nijhoff

**Description**: The term, Integrable Systems refers to a wide class of very special models, described by nonlinear differential (in the continuous case) or difference (in the discrete case) equations possessing a number of remarkable properties. One of the outstanding features is that these equations are exactly solvable in the sense that, rather than having to rely on numerical techniques or approximations, these equations allow for exact (albeit highly nontrivial) methods for their solution. Examples of these are the well-known soliton solutions of certain partial differential equations in this class. In the discrete case, the theory behind these model difference equations has been steadily developing, mostly from the early 1990s onward, together with the mathematical theories which had to be developed alongside (as they were largely non-existent in the discrete case).

The project focuses on elliptic integrable systems, which are those cases where coefficients and generating quantities for these equations are given in terms of elliptic functions. The latter generalizations of trigonometric functions have a rich mathematical structure, some aspects of which are still being explored (e.g. they play a role in Fermat's last theorem), and the integrable models which are defined through them are in a sense at the top of the food chain of models: they form the richest and most general class of equations. To a large extent the solution structure of those elliptic models has yet to be unravelled, and that will form the core of the project. In the project the student will investigate specific examples of such elliptic discrete integrable systems, which will entail not only to try and apply some well-tested techniques to these more complex cases for finding explicit solutions, but also to develop some methods for generating novel examples of such systems. As motivation, these models are expected to have relevance not only for creating novel mathematics, but also potentially for finding new models of fundamental physics.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.